Problems of the day

1.

Solve the following PDE and get Maple to plot the solution:

$$\frac{\partial u(x,t)}{\partial t} + t \frac{\partial u(x,t)}{\partial x} = 0,$$

$$u(x,0) = \frac{1}{1+x^2}.$$

2.

Solve the following PDE and get Maple to plot the solution:

$$\frac{\partial u(x,t)}{\partial t} + xt \frac{\partial u(x,t)}{\partial x} = 0,$$

$$u(x,0) = \sin(x).$$

3.

Solve the following PDE and get Maple to plot the solution:

$$\frac{\partial u(x,t)}{\partial t} + u(x,t) \frac{\partial u(x,t)}{\partial x} = 0,$$

u(x,0) = x.

4.

Solve the following PDE and get Maple to plot the solution:

$$\frac{\partial u(x,t)}{\partial t} + u(x,t) \frac{\partial u(x,t)}{\partial x} = 0,$$

$$u(x,0) = \left\{\begin{array}{ll} 0 & \mbox{if } x < 0\\ \\ x & \mbox{if } x \geq 0. \end{array}\right.$$

5.

Suppose $u_0 : {\Bbb R} \to {\Bbb R}$ is continuous. Show that u₀ is non-decreasing if and only if for every choice of $x \in \Bbb R$ and t > 0 there is a unique $\xi$ satisfying $x = u_0(\xi)t + \xi$.